Subcontests
(17)Quintuplet of integers
What is the average number of pairs of consecutive integers in a randomly selected subset of 5 distinct integers chosen from the set {1,2,3,…,30}? (For example the set {1,17,18,19,30} has 2 pairs of consecutive integers.)<spanclass=′latex−bold′>(A)</span> 32<spanclass=′latex−bold′>(B)</span> 3629<spanclass=′latex−bold′>(C)</span> 65<spanclass=′latex−bold′>(D)</span> 3029<spanclass=′latex−bold′>(E)</span> 1
MAA be simping for isosceles triangles this year
What is the sum of all possible values of t between 0 and 360 such that the triangle in the coordinate plane whose vertices are (cos40∘,sin40∘),(cos60∘,sin60∘), and (cost∘,sint∘) is isosceles?<spanclass=′latex−bold′>(A)</span> 100<spanclass=′latex−bold′>(B)</span> 150<spanclass=′latex−bold′>(C)</span> 330<spanclass=′latex−bold′>(D)</span> 360<spanclass=′latex−bold′>(E)</span> 380
Double LCM
Let M be the least common multiple of all the integers 10 through 30, inclusive. Let N be the least common multiple of M, 32, 33, 34, 35, 36, 37, 38, 39, and 40. What is the value of MN?
(<spanclass=′latex−bold′>A</span>)1(<spanclass=′latex−bold′>B</span>)2(<spanclass=′latex−bold′>C</span>)37(<spanclass=′latex−bold′>D</span>)74(<spanclass=′latex−bold′>E</span>)2886 Manipulating Conjugates
Recall that the conjugate of the complex number w=a+bi, where a and b are real numbers and i=−1, is the complex number w=a−bi. For any complex number z, let f(z)=4iz. The polynomial P(z)=z4+4z3+3z2+2z+1 has four complex roots: z1, z2, z3, and z4. Let Q(z)=z4+Az3+Bz2+Cz+D be the polynomial whose roots are f(z1), f(z2), f(z3), and f(z4), where the coefficients A, B, C, and D are complex numbers. What is B+D?(<spanclass=′latex−bold′>A</span>)−304(<spanclass=′latex−bold′>B</span>)−208(<spanclass=′latex−bold′>C</span>)12i(<spanclass=′latex−bold′>D</span>)208(<spanclass=′latex−bold′>E</span>)304 friendly reminder that equilateral hexagons are not regular
In the figure, equilateral hexagon ABCDEF has three nonadjacent acute interior angles that each measure 30∘. The enclosed area of the hexagon is 63. What is the perimeter of the hexagon?
[asy]
size(6cm);
pen p=black+linewidth(1),q=black+linewidth(5);
pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)*F;
draw(C--D--E--F--A--B--cycle,p);
dot(A,q);
dot(B,q);
dot(C,q);
dot(D,q);
dot(E,q);
dot(F,q);
label("C",C,2*S);
label("D",D,2*S);
label("E",E,2*S);
label("F",F,2*dir(0));
label("A",A,2*N);
label("B",B,2*W);
[/asy]
(<spanclass=′latex−bold′>A</span>)4(<spanclass=′latex−bold′>B</span>)43(<spanclass=′latex−bold′>C</span>)12(<spanclass=′latex−bold′>D</span>)18(<spanclass=′latex−bold′>E</span>)123 Complex Roots
Suppose that P(z),Q(z), and R(z) are polynomials with real coefficients, having degrees 2, 3, and 6, respectively, and constant terms 1, 2, and 3, respectively. Let N be the number of distinct complex numbers z that satisfy the equation P(z)⋅Q(z)=R(z). What is the minimum possible value of N?<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>3<spanclass=′latex−bold′>(E)</span>5 It's A Trap!
Let ABCD be an isosceles trapezoid with BC∥AD and AB=CD. Points X and Y lie on diagonal AC with X between A and Y, as shown in the figure. Suppose ∠AXD=∠BYC=90∘, AX=3, XY=1, and YC=2. What is the area of ABCD?[asy]
size(6cm);
usepackage("mathptmx");
import geometry;
void perp(picture pic=currentpicture,
pair O, pair M, pair B, real size=5,
pen p=currentpen, filltype filltype = NoFill){
perpendicularmark(pic, M,unit(unit(O-M)+unit(B-M)),size,p,filltype);
}
pen p=black+linewidth(1),q=black+linewidth(5);
pair C=(0,0),Y=(2,0),X=(3,0),A=(6,0),B=(2,sqrt(5.6)),D=(3,-sqrt(12.6));
draw(A--B--C--D--cycle,p);
draw(A--C,p);
draw(B--Y,p);
draw(D--X,p);
dot(A,q);
dot(B,q);
dot(C,q);
dot(D,q);
dot(X,q);
dot(Y,q);
label("2",C--Y,S);
label("1",Y--X,S);
label("3",X--A,S);
label("A",A,E);
label("B",B,N);
label("C",C,W);
label("D",D,S);
label("Y",Y,sqrt(2)*NE);
label("X",X,N);
perp(B,Y,C,8,p);
perp(A,X,D,8,p);
[/asy]
<spanclass=′latex−bold′>(A)</span>15<spanclass=′latex−bold′>(B)</span>511<spanclass=′latex−bold′>(C)</span>335<spanclass=′latex−bold′>(D)</span>18<spanclass=′latex−bold′>(E)</span>77 Polynomial Problem
Let m≥5 be an odd integer, and let D(m) denote the number of quadruples (a1,a2,a3,a4) of distinct integers with 1≤ai≤m for all i such that m divides a1+a2+a3+a4. There is a polynomial
q(x)=c3x3+c2x2+c1x+c0such that D(m)=q(m) for all odd integers m≥5. What is c1?(<spanclass=′latex−bold′>A</span>)−6(<spanclass=′latex−bold′>B</span>)−1(<spanclass=′latex−bold′>C</span>)4(<spanclass=′latex−bold′>D</span>)6(<spanclass=′latex−bold′>E</span>)11 Sum of Remainders
For n a positive integer, let R(n) be the sum of the remainders when n is divided by 2, 3, 4, 5, 6, 7, 8, 9, and 10. For example, R(15)=1+0+3+0+3+1+7+6+5=26. How many two-digit positive integers n satisfy R(n)=R(n+1)?<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>3<spanclass=′latex−bold′>(E)</span>4 GCD Condition
Let a,b, and c be positive integers such that a+b+c=23 and gcd(a,b)+gcd(b,c)+gcd(c,a)=9. What is the sum of all possible distinct values of a2+b2+c2?<spanclass=′latex−bold′>(A)</span> 259<spanclass=′latex−bold′>(B)</span> 438<spanclass=′latex−bold′>(C)</span> 516<spanclass=′latex−bold′>(D)</span> 625<spanclass=′latex−bold′>(E)</span> 687Proposed by djmathman